Formulas & tables/Mathematics/Probability/Probability of the complementary event

Probability of the complementary event

P(A′)=1−P(A)P(A')=1-P(A)

Result unit: — ($\langle 0,1\rangle$)

What it describes

The probability of the complementary event completes the probability of the event to 1.

Symbols

P(A)P(A)
probability of the event
P(A′)P(A')
probability of the complementary event

Where it comes from

Instead of counting an event directly, it is sometimes easier to count that it does NOT occur and subtract from 1 — especially with "at least one".

When to use it

When the complementary event is easier to count.

Example

The probability of success is 0.3. Find the probability of failure.
SubstitutionP(A)=0.3P(A)=0.3.
ResultP(A′)=1−0.3=0.7P(A')=1-0.3=0.7.

Related formulas

Classical probability Probability of the union of events

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